Approximating the distributions of estimators of financial risk under an asymmetric Laplace law

Explicit expressions are derived for parametric and nonparametric estimators (NPEs) of two measures of financial risk, value-at-risk (VaR) and conditional value-at-risk (CVaR), under random sampling from the asymmetric Laplace (AL) distribution. Asymptotic distributions are established under very general conditions. Finite sample distributions are investigated by means of saddlepoint approximations. The latter are highly computationally intensive, requiring novel approaches to approximate moments and special functions that arise in the evaluation of the moment generating functions. Plots of the resulting density functions shed new light on the quality of the estimators. Calculations for CVaR reveal that the NPE enjoys greater asymptotic efficiency relative to the parametric estimator than is the case for VaR. An application of the methodology in modeling currency exchange rates suggests that the AL distribution is successful in capturing the peakedness, leptokurticity, and skewness, inherent in such data. A demonstrated superiority in the resulting parametric-based inferences delivers an important message to the practitioner.